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Technical Briefs

Elastoplastic Stress Analysis and Residual Stresses in Cylindrical Bar Under Combined Bending and Torsion

[+] Author and Article Information
V. Kobelev

Department of Mechanical Engineering, Faculty IV,  University of Siegen, Paul-Bonatz-Str. 9-11, D-57076, Siegen, Germanykobelev@imr.mb.uni-siegen.de

J. Manuf. Sci. Eng 133(4), 044502 (Aug 11, 2011) (11 pages) doi:10.1115/1.4004496 History: Received February 03, 2009; Revised June 14, 2011; Published August 11, 2011; Online August 11, 2011

The excessive stresses during the coiling of helical springs could lead to breakage of the rod. Moreover, the high level of residual stress in the formed helical spring reduces considerably its fatigue life. For the practical estimation of residual and coiling stresses in the helical springs the analytical formulas are necessary. In this paper the analytical solution of the problem of elastic–plastic deformation of cylindrical bar under combined bending and torsion moments is found for a special nonlinear stress–strain law. The obtained solution allows the analysis of the active stresses during the combined bending and twist. Moreover, the residual stresses in the bar after springback are also derived in closed analytical form. The results of this analysis are applied to the actual engineering problem of determination of stresses during the manufacturing of helical coiled springs. A practically important example, describing the manufacturing of helical coiled spring is worked out to illustrate the simplicity achieved in determining the plasticization process and residual stresses. The obtained results match the reported measured values. The developed method does not require numerical simulation and is perfectly suited for programming of coiling machines, estimation of loads during manufacturing of cold-wounded helical springs and for dimensioning and wear calculation of coiling tools.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Modified Ramberg–Osgood law for negative values of k (k=  1 and k=  1/2)

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Figure 2

Modified Ramberg–Osgood law for positive values of k (k=  1 and k=  1/2)

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Figure 3

Cylindrical bar under combined bending and torsion

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Figure 4

Bending moment and torque as a function of loading parameter α

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Figure 5

Shear stress in the cross-section in maximal plasticization state

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Figure 6

Axial normal stress in the cross-section in maximal plasticization state

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Figure 7

Equivalent stress in the cross-section in maximal plasticization state

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Figure 8

Plastic stresses on the contour of the cross-section in maximal plasticization state

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Figure 9

Residual equivalent stress in the unloaded state

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Figure 10

Residual shear stress in the unloaded state

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Figure 11

Residual axial normal stress in the unloaded state

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Figure 12

Residual stresses on the contour of the cross-section in the unloaded state

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Figure 13

Residual stresses along the positive axis 0 < x < r in the unloaded state

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